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	<div align="center"><h1>Stochastic volatility models for financial time series</h1></div>
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                     Code: <a href="sdv.tpl">sdv.tpl</a><br>
                     Data: 		<a href="sdv.dat">sdv.dat</a><br>
                     Initial values: <a href="sdv.pin">sdv.pin</a><br>
                     All required files (DOS): <a href="sdv.zip">sdv.zip</a><br>
                     All required files (linux): <a href="sdv.tar.gz">sdv.tar.gz</a><br>
                     Results: <a href="sdv.par">sdv.par</a><br><br>
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                     In a <a href="../admb_tutorial.html">DOS</a> window<br> 
					 Under <a href="../admb_tutorial.html">linux</a><br>
                     Command line option: <TT>-ilmn 5</TT>.
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             <font face="Arial, Helvetica" color="White"><b>Results: Computation times</b></font>
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                     ADMB-RE: 22 seconds.<br>
					 BUGS : 1.7 hours ; (<A HREF="../citations.html#meye:yu:2000">Meyer &amp; Yu, 2000, Table 1</A>).<br>
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<h3><strong>Model description</strong></h3>
Stochastic volatility models are used in mathematical finance to describe the evolution of asset returns, 
which typically exhibit changing variances over time. As an illustration we use a dataset 
previously analyzed by <A HREF="../citations.html#harv:ruiz:shep:1994">Harvey et al. (1994)</A>, and later by several other authors. 
The data consist of a time series of daily pound/dollar exchange rates {<em>z<sub>t</sub></em>}  from the period 01/10/81 to 28/6/85. 
The series of interest are the daily mean-corrected returns  {<em>y<sub>t</sub></em>}, given by the transformation
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<div align="center"><em>y<sub>t</sub></em> = log(<em>z<sub>t</sub></em>)-log(<em>z<sub>t-1</sub></em>)
- average[log<em>z<sub>i</sub></em>-log<em>z<sub>i-1</sub></em>].
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The stochastic volatility model allows the variance of <em>y<sub>t</sub></em> to vary smoothly with time. 
This is achieved by assuming that <em>y<sub>t</sub></em> ~ N(0,<em><FONT FACE="Symbol">s</FONT><sub>t</sub></em>), 
where <em><FONT FACE="Symbol">s</FONT><sub>t</sub></em> = exp{-0.5(<em><FONT FACE="Symbol">m</FONT><sub>x</sub></em>+<em>x<sub>t</sub></em>)}. 
Here, the smoothly varying component {<em>x<sub>t</sub></em>} is assumed to follow an autoregression
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<em>x<sub>t</sub></em> = <FONT FACE="Symbol">b</FONT><em>x<sub>t-1</sub></em> + <FONT FACE="Symbol">e</FONT><sub>t</sub>,
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where <FONT FACE="Symbol">e</FONT><sub>t</sub> ~ N(0,<em><FONT FACE="Symbol">s</FONT></em><sup>2</sup>). 
Further details about the model can be found here: <a href="sdv.pdf">sdv.pdf</a>.

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